Task 1 - Exploratory analysis of the data¶
Heart Disease Dataset¶
Team composition:¶
- Amihaesei Sergiu
- Stoica George
Preliminaries¶
In [94]:
import pandas as pd
import numpy as np
import plotly.graph_objects as go
import plotly.express as px
import plotly.io as pio
import matplotlib.pyplot as plt
import scipy.stats as stats
from sklearn.decomposition import PCA
from sklearn.manifold import TSNE
import seaborn as sns
!pip install umap-learn
!pip install projection-pursuit
import umap
from skpp import ProjectionPursuitRegressor
#When saving to HTML uncomment the line below
pio.renderers.default = "notebook" #"colab" is default
In [95]:
data = pd.read_csv("./heart.csv")
Checking data consistency:
In [96]:
data.isna().any()
Out[96]:
Univariate Analysis¶
Central tendence & Spread¶
Mode - represents the most frequent value. Since it's not included in Pandas' discribe, we had to call it separately. It returns multiple rows if there are multiple modes available, hence we only took the first row.
In [97]:
numerical_data = data[["age", "cp", "trestbps", "chol", "fbs", "thalach"]]
numerical_data.mode().iloc[0]
Out[97]:
Describe offers descriptive statistics: the mean, median (here represented by 50%), range and standard deviation
In [98]:
numerical_data.describe()
Out[98]:
Form of the distribution¶
Skewness measures the asymmetry of a distribution:
- When the value of the skewness if zero, the distribution follows the normal distribution
- When the value of the skewness is negative, the tail of the distribution is longer towards the left hand side of the curve.
- When the value of the skewness is positive, the tail of the distribution is longer towards the right hand side of the curve
From the dataset, fbs has the highest positive value, while sex has the highest negative value.
The variables sex is really skewed to the right, and since it is categorical it means there are more males than females in the dataset. The variable age is really close to 0, which implies that it almost follows the normal distribution. For the rest of the variables, the skewness is more visible in the density plot.
In [99]:
data.skew()
Out[99]:
Kurtosis measures the peakedness of a distribution:
- High kurtosis in a data set is an indicator that data has heavy outliers.
- Low kurtosis in a data set is an indicator that data has lack of outliers.
From the dataset variables, chol has heavy outliers, while thalach lacks outliers
The variable chol, which is the level of cholesterol, has heavy outliers. The variables thalach, thal, and age dont have that many outliers.
In [100]:
data.kurtosis()
Out[100]:
Frequencies for categorical attributes¶
We can see that indeed the sex is heavily disproportionate, aswell as fbs, exang. Some contain even some sort of outliers like restecg, ca, thal.
In [101]:
categorical_variables = ["sex", "fbs", "restecg", "exang", "cp", "slope", "ca", "thal"]
for i in categorical_variables:
print(data[i].value_counts())
Graphs¶
Density¶
Here we can clearly see how skewness and kurtosis were determined
- We can observe that there are more men than women, the majority of the subjects are between 50 and 60 years old.
In [102]:
nrow = 7
ncol = 2
fig, axes = plt.subplots(nrow, ncol, constrained_layout=True)
count = 0
for r in range(nrow):
for c in range(ncol):
title = 'Attribute "' + data.columns[count] + '" density plot'
data.iloc[:, count].plot.density(ax=axes[r,c], title=title, figsize=(10,20))
count += 1
Frequency¶
Shows how the attributes' values are distributed
In [103]:
nrow = 7
ncol = 2
fig, axes = plt.subplots(nrow, ncol, constrained_layout=True)
count = 0
for r in range(nrow):
for c in range(ncol):
title = 'Attribute "' + data.columns[count] + '" frequency plot'
data.iloc[:, count].plot.hist(ax=axes[r,c], title=title, figsize=(10,20))
count += 1
Boxplot¶
The boxplot shows the minimum, first quartile (25% of the data), median (50% of the data), third quartile (75% of the data) and the maximum. The values outside of $[minimum, maximum]$ are plotted as points and are to be considered outliers.
In [104]:
nrow = 7
ncol = 2
fig, axes = plt.subplots(nrow, ncol, constrained_layout=True)
count = 0
for r in range(nrow):
for c in range(ncol):
title = 'Attribute "' + data.columns[count] + '" boxplot'
data.iloc[:, count].plot.box(ax=axes[r,c], title=title, figsize=(10,20))
count += 1
Conclusion¶
- Attributes thal, ca, oldpeak, thalach, fbs, chol, trestbps have outliers
- Categorical attributes are not uniformely distributed
Bivariate / Multivariate Analysis¶
Pearson correlation¶
- Measures the strenght of the linear dependency between two variables.
- Values closer to 1 indicate a direct correlation between attributes.
- Values closer to -1 indicate an inverse correlation between attributes.
- Values closer to 0 indicate no correlation between attributes.
In [105]:
numerical_data.corr(method="pearson").style.background_gradient(cmap='Blues')
Out[105]:
Spearman correlation¶
- Unlike Pearson, works with monotonic relationships as well.
In [106]:
numerical_data.corr(method="spearman").style.background_gradient(cmap='Blues')
Out[106]:
Kendall correlation¶
- Has a smaller gross error sensitivity and a smaller asymptotic variance than Spearman
In [107]:
numerical_data.corr(method="kendall").style.background_gradient(cmap='Blues')
Out[107]:
From the above correlation matrices we can observe a small correlation between thalach and cp and a small inverse correlation between thalach and age.¶
Independence tests¶
Chi-Square Test for Independence¶
- Used to examine the differences between categorical variables in order to evaluate the difference between the predicted and observed results
- The p-value tells whether the test is significant or not
- A p-value over 0.05 shows that the test is not significant enough
- The ${chi}^2$ represents the expected frequencies.
In [108]:
dof = 0.05
categorical_variables = ["sex", "fbs", "restecg", "exang", "cp", "slope", "ca", "thal", "target"]
semnificative = []
for i in range(len(categorical_variables)):
for j in range(i + 1, len(categorical_variables)):
ct = pd.crosstab(data[categorical_variables[i]], data[categorical_variables[j]], margins=True)
values = [ct[i][0:len(ct.index) - 1].values for i in range(len(ct.columns) - 1)]
chi2, p, _, _ = stats.chi2_contingency(values)
print(f"Independence between {categorical_variables[i]:<7} and {categorical_variables[j]:<7}. Chi-squared test: ")
print(f"chi2: {chi2:.3f}, p-value: {p:.3f}")
print(f"{'Failed, p-value bigger than ' + str(dof) if p > dof else 'Relevant'}\n")
if (p < 0.0001):
semnificative.append([categorical_variables[i], categorical_variables[j]])
print("Semnificative pairs")
for a, b in semnificative:
print(a, b)
Fisher's Test¶
- Used in the analysis of contingency tables.
- It works only for binary variables.
- The p-values ilustrate the significance of the test.
- The ratio is the prior odds ratio.
In [109]:
dof = 0.05
binary_variables = ["sex", "fbs", "exang", "target"]
semnificative = []
for i in range(len(binary_variables)):
for j in range(i + 1, len(binary_variables)):
ct = pd.crosstab(data[binary_variables[i]], data[binary_variables[j]], margins=True)
values = [ct[i][0:len(ct.index) - 1].values for i in range(len(ct.columns) - 1)]
ratio, p2 = stats.fisher_exact(values)
print(f"Independence between {binary_variables[i]:<7} and {binary_variables[j]:<7}. Fisher's exact test:")
print(f"ratio: {ratio:.3f}, p-value: {p2:.3f}")
print(f"{'Failed, p-value bigger than ' + str(dof) if p2 > dof else 'Relevant'}\n")
if (p2 < 0.0001):
semnificative.append([binary_variables[i], binary_variables[j]])
print("Semnificative pairs")
for a, b in semnificative:
print(a, b)
T-test Independence¶
- Mostly used for identifying a statistically significant difference in the means between 2 groups.
- The t-values quantify the difference between the arithmetic means of the two samples.
- We split each variable in the dataset into 2 groups, split according to each binary variable
In [110]:
dof = 0.05
binary_variables = ["sex", "fbs", "exang", "target"]
semnificative = []
for column in numerical_data.columns:
for attribute in binary_variables:
if column != attribute:
group_1 = data[column][data[attribute] == 1]
group_2 = data[column][data[attribute] == 0]
t, p = stats.ttest_ind(group_1, group_2)
print(f"Variable {column:<8} separated by {attribute:<6} has t-value: {t:.3f} and p-value: {p:.3f}.")
print(f"{'Failed, p-value bigger than ' + str(dof) if p > dof else 'Relevant'}\n")
if (p < 0.0001):
semnificative.append([column, attribute])
print("Semnificative pairs")
for a, b in semnificative:
print(a, b)
Vizualization¶
Scatterplot on 2-variable combinations¶
If two variables are correlated we should see a relateness between the points.
In [111]:
fig = go.Figure()
axis_labels = []
count = 0
for i in range(len(data.columns)):
for j in range(i + 1, len(data.columns)):
count += 1
fig2 = px.scatter(data, x=data.iloc[:, i],
y=data.iloc[:, j],
color="target")
fig.add_trace(fig2.data[0])
axis_labels.append([data.columns[i], data.columns[j]])
fig.data[count - 1].visible = True
steps = []
for i in range(len(fig.data)):
step = dict(
method="update",
args=[{"visible": [False] * len(fig.data)},
{"title": "Slider switched to step: " + str(i),
"xaxis.title": axis_labels[i][0],
"yaxis.title": axis_labels[i][1]
}],
)
step["args"][0]["visible"][i] = True
steps.append(step)
sliders = [dict(
active=count - 1,
currentvalue={"prefix": "Frequency: "},
pad={"t": 50},
steps=steps
)]
fig.update_layout(
sliders=sliders,
plot_bgcolor='rgba(100,100,100,2)'
)
fig.show()
3D Graphs¶
In [112]:
variables = ["age", "trestbps", "chol", "thalach", "oldpeak"]
axis_labels = []
fig = go.Figure()
count = 0
for i in range(len(variables)):
for j in range(i + 1, len(variables)):
for k in range(j + 1, len(variables)):
count += 1
fig2 = px.scatter_3d(data, x=variables[i],
y=variables[j],
z=variables[k],
color="target")
fig.add_trace(fig2.data[0])
axis_labels.append([variables[i], variables[j], variables[k]])
fig.data[count - 1].visible = True
steps = []
for i in range(len(fig.data)):
step = dict(
method="update",
args=[{"visible": [False] * len(fig.data)},
{"title": "Slider switched to step: " + str(i),
"scene.xaxis.title": axis_labels[i][0],
"scene.yaxis.title": axis_labels[i][1],
"scene.zaxis.title": axis_labels[i][2],
}],
)
step["args"][0]["visible"][i] = True
steps.append(step)
sliders = [dict(
active=count - 1,
currentvalue={"prefix": "Frequency: "},
pad={"t": 50},
steps=steps
)]
fig.update_layout(
sliders=sliders
)
fig.show()
Principal Components Analysis¶
- PCA is used in order to reduce dimensionality, increase interpretability while also minimizing information loss.
- Using the first 3 principal components we would explain more than 40% of the variance.
In [113]:
normalized_data = data.copy()
normalized_data = normalized_data.drop(columns=["target"])
normalized_data = (normalized_data - normalized_data.mean()) / normalized_data.std()
pca = PCA(n_components=normalized_data.shape[1])
pca.fit(normalized_data)
%matplotlib inline
plt.plot(pca.explained_variance_ratio_)
plt.ylabel('Explained Variance')
plt.xlabel('Components')
plt.xticks([i for i in range(len(pca.explained_variance_ratio_))])
plt.show()
Using 3 principal components:¶
In [114]:
pca3 = PCA(n_components=3)
new_data_3 = pca3.fit_transform(normalized_data)
new_data_3 = pd.DataFrame(data = new_data_3, columns = ['PCA 1', 'PCA 2', 'PCA 3'])
print(f'Explained variation per principal component: {pca3.explained_variance_ratio_}. Total: {sum(pca3.explained_variance_ratio_)}')
pca3_by_variables = pd.DataFrame(pca3.components_, columns = data.drop(columns="target").columns)
How much does each variable contribute to the principal components¶
In [115]:
pca3_by_variables.T.index
Out[115]:
Using 5 principal components¶
In [116]:
pca5 = PCA(n_components=5)
new_data_5 = pca5.fit_transform(normalized_data)
new_data_5 = pd.DataFrame(data = new_data_5, columns = ['PCA 1', 'PCA 2', 'PCA 3', 'PCA 4', 'PCA 5'])
print(f'Explained variation per principal component: {pca5.explained_variance_ratio_}. Total: {sum(pca5.explained_variance_ratio_)}')
pca_by_variables = pd.DataFrame(pca5.components_, columns = data.drop(columns="target").columns)
How much does each variable contribute to the principal components¶
In [117]:
pca_by_variables
Out[117]:
In [118]:
new_data_3["target"] = data["target"]
fig = px.scatter_3d(new_data_3, x=new_data_3.columns[0], y=new_data_3.columns[1], z=new_data_3.columns[2], color='target')
for attribute in pca3_by_variables.T.index:
x = np.array([0, pca3_by_variables[attribute][0]]) * 10
y = np.array([0, pca3_by_variables[attribute][1]]) * 10
z = np.array([0, pca3_by_variables[attribute][2]]) * 10
fig.add_trace(go.Scatter3d(x=x, y=y,z=z, mode='lines', name=attribute))
fig.update_layout(legend=dict(
orientation="h",
yanchor="bottom",
y=1.02,
xanchor="right",
x=1
))
fig.show()
Non-linear bi-dimensional mappings¶
Non-linear mappings transforms the data from a multi-dimensional space to a bi-dimensional one.
In [119]:
def sammon(x, n, display = 2, inputdist = 'raw', maxhalves = 20, maxiter = 500, tolfun = 1e-9, init = 'default'):
import numpy as np
from scipy.spatial.distance import cdist
if inputdist == 'distance':
D = x
if init == 'default':
init = 'cmdscale'
else:
D = cdist(x, x)
if init == 'default':
init = 'pca'
if inputdist == 'distance' and init == 'pca':
raise ValueError("Cannot use init == 'pca' when inputdist == 'distance'")
if np.count_nonzero(np.diagonal(D)) > 0:
raise ValueError("The diagonal of the dissimilarity matrix must be zero")
N = x.shape[0]
scale = 0.5 / D.sum()
D = D + np.eye(N)
print(np.count_nonzero(D<=0))
if np.count_nonzero(D<=0) > 0:
raise ValueError("Off-diagonal dissimilarities must be strictly positive")
Dinv = 1 / D
if init == 'pca':
[UU,DD,_] = np.linalg.svd(x)
y = UU[:,:n]*DD[:n]
else:
y = np.random.normal(0.0,1.0,[N,n])
one = np.ones([N,n])
d = cdist(y,y) + np.eye(N)
dinv = 1. / d
delta = D-d
E = ((delta**2)*Dinv).sum()
for i in range(maxiter):
delta = dinv - Dinv
deltaone = np.dot(delta,one)
g = np.dot(delta,y) - (y * deltaone)
dinv3 = dinv ** 3
y2 = y ** 2
H = np.dot(dinv3,y2) - deltaone - np.dot(2,y) * np.dot(dinv3,y) + y2 * np.dot(dinv3,one)
s = -g.flatten(order='F') / np.abs(H.flatten(order='F'))
y_old = y
for j in range(maxhalves):
s_reshape = np.reshape(s, (-1,n),order='F')
y = y_old + s_reshape
d = cdist(y, y) + np.eye(N)
dinv = 1 / d
delta = D - d
E_new = ((delta**2)*Dinv).sum()
if E_new < E:
break
else:
s = 0.5*s
if j == maxhalves-1:
print('Warning: maxhalves exceeded. Sammon mapping may not converge...')
if abs((E - E_new) / E) < tolfun:
if display:
print('TolFun exceeded: Optimisation terminated')
break
E = E_new
if display > 1:
print('epoch = %d : E = %12.10f'% (i+1, E * scale))
if i == maxiter-1:
print('Warning: maxiter exceeded. Sammon mapping may not have converged...')
E = E * scale
return [y,E]
In [120]:
data_matrix = data.to_numpy()[:, :-1]
x, index = np.unique(data_matrix, axis=0, return_index=True)
target = data.to_numpy()[:, -1]
target = target[index]
y, E = sammon(x, 2)
fig = go.Figure()
fig.add_trace(go.Scatter(x=y[target==0, 0], y=y[target==0, 1],
mode='markers',
name='Negative Heart Disease'))
fig.add_trace(go.Scatter(x=y[target==1, 0], y=y[target==1, 1],
mode='markers',
name='Positive Heart Disease'))
fig.update_layout(title='Sammon Mapping')
fig.show()
t-SNE¶
In [121]:
y = TSNE(n_components=2, learning_rate='auto', init="pca").fit_transform(x)
fig = go.Figure()
fig.add_trace(go.Scatter(x=y[target==0, 0], y=y[target==0, 1],
mode='markers',
name='Negative Heart Disease'))
fig.add_trace(go.Scatter(x=y[target==1, 0], y=y[target==1, 1],
mode='markers',
name='Positive Heart Disease'))
fig.update_layout(title='t-SNE Mapping')
fig.show()
uMap¶
In [122]:
# Has learning_rate, different distance metrics, spread, min_dist
y = umap.UMAP().fit_transform(x)
fig = go.Figure()
fig.add_trace(go.Scatter(x=y[target==0, 0], y=y[target==0, 1],
mode='markers',
name='Negative Heart Disease'))
fig.add_trace(go.Scatter(x=y[target==1, 0], y=y[target==1, 1],
mode='markers',
name='Positive Heart Disease'))
fig.update_layout(title='uMap Mapping')
fig.show()
Projection Pursuit¶
Available in the other document
Conditional boxplots¶
Plotting every numerical variable according to a categorical one as a boxplot. If the boxplots notches are not overlapping that means that there is a statistically significant difference between the medians.
In [123]:
numerical_variables = ["age", "trestbps", "chol", "thalach", "oldpeak"]
categorical_variables = ["sex", "fbs", "restecg", "exang", "cp", "slope", "ca", "thal", "target"]
axis_labels = []
fig = go.Figure()
count = 0
for nv in numerical_variables:
for cv in categorical_variables:
count += 1
axis_labels.append([cv, nv])
fig.add_trace(go.Box(x=data[cv], y=data[nv]))
fig.data[count - 1].visible = True
steps = []
for i in range(len(fig.data)):
step = dict(
method="update",
args=[{"visible": [False] * len(fig.data)},
{"title": "Slider switched to step: " + str(i),
"xaxis.title": axis_labels[i][0],
"yaxis.title": axis_labels[i][1]
}],
)
step["args"][0]["visible"][i] = True
steps.append(step)
sliders = [dict(
active=count - 1,
currentvalue={"prefix": "Frequency: "},
pad={"t": 50},
steps=steps
)]
fig.update_layout(
sliders=sliders
)
fig.show()
We can see that:
- age medians according to sex, fb, exang are (almost) overlapping.
- trestbps -//- to sex, exang, slope, target
- chol -//- to sex, fbs, exang, cp, slope
- thalach -//- to fbs
- oldpeak -//- to sex, fbs
Overlay Histogram¶
Allows us to visualize and compare the decompositions of the numeric value superimposed on each other.
In [124]:
numerical_variables = ["age", "trestbps", "chol", "thalach", "oldpeak"]
categorical_variables = ["sex", "fbs", "restecg", "exang", "cp", "slope", "ca", "thal", "target"]
axis_labels = []
show_traces = []
fig = go.Figure()
count = 0
for nv in numerical_variables:
for cv in categorical_variables:
fig2 = px.histogram(data, x=nv, color=cv)
traces_to_show = []
for i in range(len(fig2.data)):
traces_to_show.append(count)
count += 1
histogram = fig2.data[i]
histogram.name = cv + " " + str(histogram.name)
histogram.opacity = 0.8
fig.add_trace(histogram)
show_traces.append(traces_to_show)
axis_labels.append([nv, cv])
steps = []
for i in range(len(show_traces)):
step = dict(
method="update",
args=[{"visible": [False] * len(fig.data)},
{"title": "Showing in relation to " + axis_labels[i][1],
"xaxis.title": axis_labels[i][0],
"yaxis.title": "count"
}],
)
for j in show_traces[i]:
step["args"][0]["visible"][j] = True
steps.append(step)
sliders = [dict(
active=count - 1,
currentvalue={"prefix": "Frequency: "},
pad={"t": 50},
steps=steps
)]
fig.update_layout(
sliders=sliders,
barmode="overlay"
)
fig.show()
Stacked Histogram¶
Used to determine the relative decomposition of an attribute according to a categorical variable. Thus, we can see the contribution of each secondary group in overall value.
In [125]:
numerical_variables = ["age", "trestbps", "chol", "thalach", "oldpeak"]
categorical_variables = ["sex", "fbs", "restecg", "exang", "cp", "slope", "ca", "thal", "target"]
axis_labels = []
show_traces = []
fig = go.Figure()
count = 0
for nv in numerical_variables:
for cv in categorical_variables:
fig2 = px.histogram(data, x=nv, color=cv)
traces_to_show = []
for i in range(len(fig2.data)):
traces_to_show.append(count)
count += 1
histogram = fig2.data[i]
histogram.name = cv + " " + str(histogram.name)
histogram.opacity = 0.8
fig.add_trace(histogram)
show_traces.append(traces_to_show)
axis_labels.append([nv, cv])
steps = []
for i in range(len(show_traces)):
step = dict(
method="update",
args=[{"visible": [False] * len(fig.data)},
{"title": "Showing in relation to " + axis_labels[i][1],
"xaxis.title": axis_labels[i][0],
"yaxis.title": "count"
}],
)
for j in show_traces[i]:
step["args"][0]["visible"][j] = True
steps.append(step)
sliders = [dict(
active=count - 1,
currentvalue={"prefix": "Frequency: "},
pad={"t": 50},
steps=steps
)]
fig.update_layout(
sliders=sliders,
barmode="stack"
)
fig.show()
Dependency analysis¶
Conditioned Boxplots¶
Helps us visualize how dependent an attribute is to the target.
In [126]:
axis_labels = []
fig = go.Figure()
count = 0
for pred in data.columns[:-1]:
count += 1
axis_labels.append(["target", pred])
fig.add_trace(go.Box(x=data["target"], y=data[pred]))
fig.data[count - 1].visible = True
steps = []
for i in range(len(fig.data)):
step = dict(
method="update",
args=[{"visible": [False] * len(fig.data)},
{"title": "Slider switched to step: " + str(i),
"xaxis.title": axis_labels[i][0],
"yaxis.title": axis_labels[i][1]
}],
)
step["args"][0]["visible"][i] = True
steps.append(step)
sliders = [dict(
active=count - 1,
currentvalue={"prefix": "Frequency: "},
pad={"t": 50},
steps=steps
)]
fig.update_layout(
sliders=sliders
)
fig.show()
Correlation between attributes and target¶
In [127]:
data.corr(method="kendall")[-1:].style.background_gradient(cmap='Blues')
Out[127]:
In [128]:
corr_data = data.corr(method="kendall")[-1:].to_numpy()[0,:-1]
fig = go.Figure()
fig.add_trace(go.Bar(x=data.columns[:-1], y=corr_data))
fig.update_layout(title="Correlation barplot")
fig.show()
In [129]:
dof = 0.05
categorical_variables = ["sex", "fbs", "restecg", "exang", "cp", "slope", "ca", "thal"]
for i in range(len(categorical_variables)):
ct = pd.crosstab(data[categorical_variables[i]], data["target"], margins=True)
values = [ct[i][0:len(ct.index) - 1].values for i in range(len(ct.columns) - 1)]
chi2, p, _, _ = stats.chi2_contingency(values)
print(f"Independence between {categorical_variables[i]:<7} and target. Chi-squared test: ")
print(f"chi2: {chi2:.3f}, p-value: {p:.3f}")
print(f"{'Failed, p-value bigger than ' + str(dof) if p > dof else 'Relevant'}\n")
Fisher's test¶
- We can see that sex and exang test results are relevant while fbs test results are not.
- The ratio is the prior odds ratio.
In [130]:
dof = 0.05
binary_variables = ["sex", "fbs", "exang"]
for i in range(len(binary_variables)):
ct = pd.crosstab(data[binary_variables[i]], data["target"], margins=True)
values = [ct[i][0:len(ct.index) - 1].values for i in range(len(ct.columns) - 1)]
ratio, p2 = stats.fisher_exact(values)
print(f"Independence between {binary_variables[i]:<7} and target. Fisher's exact test:")
print(f"ratio: {ratio:.3f}, p-value: {p2:.3f}")
print(f"{'Failed, p-value bigger than ' + str(dof) if p2 > dof else 'Relevant'}\n")
T-test¶
- We can see that t-values obtained by using sex and exang are relevant, while thos obtained by using fbs are not.
- The t-values quantify the difference between the arithmetic means of the two samples
In [131]:
binary_variables = ["sex", "fbs", "exang"]
for column in binary_variables:
group_1 = data["target"][data[column] == 1]
group_2 = data["target"][data[column] == 0]
t, p = stats.ttest_ind(group_1, group_2)
print(f"Variable target separated by {column:<5} has t-value: {t:.3f} and p-value: {p:.3f}.")
print(f"{'Failed, p-value bigger than 0.05' if p > 0.05 else 'Relevant'}\n")
Conclusion¶
- From binary variables, the variables more likely to influence the target are:
- exang, according to all three tests
- sex, according to all three tests
- From the other categorial variables, the variables more likely to influence the target are:
- cp, according to Chi-squared Test
- ca, according to Chi-squared Test
- thal, according to Chi-squared Test
- slope, according to Chi-squared Test
- From the other variables, the ones more likely to influence the target are:
- oldpeak, according to the correlation matrix and barplots
- thalach, according to the correlation matrix and barplots